Let be the given function. Of course, this is the equation of a parabolid with its tip - or minimum - at the origin. However, let be the constraint equation, meaning that we don't search for the global minimum but for the minimum along the line . There are two ways to go about it. The simple but inconvenient way is to substitute in , thus rendering a function of only. Equating the derivative to zero we find the locus of the conditional minimum, and . The process of substitution is, in general, tedious.
A more elegant method is this: defining a (undetermined) Lagrange
find the minimum of the function